Rank Covariance Methods for the Analysis of Survival Data

A procedure for comparing survival times between several groups of patients through rank analysis of covariance was introduced by WOOLSON and LACHENBRUCH (1983). It is a modification of Quade' rank analysis of covariance procedure (1967) and can be used for the analysis of right-censored data. In this paper, two additional modifications of Quade' original test statistic are proposed and compared to the original modification introduced by Woolson and Lachenbruch. These statistics are compared to one another and to the score test from Cox' proportional hazards model by way of a limited Monte Carlo study. One of the statistics, Q_R2, is recommended for general use for the rank analysis of covariance of right-censored survivorship data.     

Estimating Location Parameters Using Censored Rank Statistics

In this paper censored data rank location estimators are obtained by using censored one-sample rank test statistics of the location parameter and censored two-sample rank test statistics of the shift of location parameter. Also, methods for constructing censored small sample confidence intervals and asymptotic confidence intervals for the location are considered. Generalizations of the solutions from uncensored one-sample and two-sample rank tests are utilized.     

Comparison between Determination of Free Thyroxine Index and Effective Thyroxine Ratio in Human Serum

Four thyroid function indices were determined in the sera of 181 patients: total thyroxine (T-4), the triiodothyronine resin uptake (T-3 B.C.-index), the free thyroxine (T-4) index, and the effective thyroxine ratio (E.T.R.).A statistically significant correlation between the free T-4 index and E.T.R. could be established though this correlation seemed to be worse than could be expected from the literature. Two sets of correlation coefficients were shown: one obtained with the ordinarily used Pearson or product-moment correlation and one with the Spearman rank correlation. Because the assumptions and requirements for the proper use of the product-moment correlation statistics, such as a normal distribution of both variables, are seldom met, expecially not if a selected number of controls and patients is used for the correlation, it is advisable to use some kind of rank correlation statistics instead.The advantages and disadvantages of the E.T.R. and free T-4 index determination are described.     

Maximum of the weighted Kaplan-Meier tests with application to cancer prevention and screening trials

A class of maximum weighted Kaplan-Meier test statistics is described where the weight functions are chosen from a family of smooth functions. The investigated test statistic is robust and sensitive to a variety of alternatives that are often observed in cancer prevention and screening trials. A simulation study is performed to compare the size and power properties between the proposed test statistics and some existing ones. We illustrate the procedure using data from a clinical trial of a breast cancer screening program.     

Generalized maximally selected statistics

SUMMARY: Maximally selected statistics for the estimation of simple cutpoint models are embedded into a generalized conceptual framework based on conditional inference procedures. This powerful framework contains most of the published procedures in this area as special cases, such as maximally selected chi(2) and rank statistics, but also allows for direct construction of new test procedures for less standard test problems. As an application, a novel maximally selected rank statistic is derived from this framework for a censored response partitioned with respect to two ordered categorical covariates and potential interactions. This new test is employed to search for a high-risk group of rectal cancer patients treated with a neo-adjuvant chemoradiotherapy. Moreover, a new efficient algorithm for the evaluation of the asymptotic distribution for a large class of maximally selected statistics is given enabling the fast evaluation of a large number of cutpoints.     

Testing equality of survival functions based on both paired and unpaired censored data

We introduce two test procedures for comparing two survival distributions on the basis of randomly right-censored data consisting of both paired and unpaired observations. Our procedures are based on generalizations of a pooled rank test statistic previously proposed for uncensored data. One generalization adapts the Prentice-Wilcoxon score, while the other adapts the Akritas score. The use of these particular scoring systems in pooled rank tests with randomly right-censored paired data has been advocated by several researchers. Our test procedures utilize the permutation distributions of the test statistics based on a novel manner of permuting the scores. Permutation versions of tests for right-censored paired data and for two independent right-censored samples that use the proposed scoring systems are obtained as special cases of our test procedures. Simulation results show that our test procedures have high power for detecting scale and location shifts in exponential and log-logistic distributions for the survival times. We also demonstrate the advantages of our test procedures in terms of utilizing randomly occurring unpaired observations that are discarded in test procedures for paired data. The tests are applied to skin graft data previously reported elsewhere.     

Maximally Selected Rank Statistics for Dose‐Response Problems

We consider the bivariate situation of some quantitative, ordinal, binary or censored response variable and some quantitative or ordinal exposure variable (dose) with a hypothetical effect on the response. Data can either be the outcome of a planned dose‐response experiment with only few dose levels or of an observational study where, for example, both exposure and response variable are observed within each individual. We are interested in testing the null hypothesis of no effect of the dose variable vs. a dose‐response function depending on an unknown ‘threshold’ parameter. The variety of dose‐response functions considered ranges from no observed effect level (NOEL) models to umbrella alternatives. Here we discuss generalizations of the method of Lausen & Schumacher (Biometrics, 1992, 48, 73–85)which are based on combinations of two‐sample rank statistics and rank statistics for trend. Our approach may be seen as a generalization of a proposal for change‐point problems. Using the approach of Davies (Biometrika, 1987, 74, 33–43)we derive and approximate the asymptotic null distribution for a large number of thresholds considered. We use an improved Bonferroni inequality as approximation for a small number of thresholds considered. Moreover, we analyse the small sample behaviour by means of a Monte‐Carlo study. Our paper is illustrated by examples from clinical research and epidemiology.     

Assessing the robustness of randomization tests: examples from behavioural studies

Behavioural studies are commonly plagued with data that violate the assumptions of parametric statistics. Consequently, classic nonparametric methods (e.g. rank tests) and novel distribution-free methods (e.g. randomization tests) have been used to a great extent by behaviourists. However, the robustness of such methods in terms of statistical power and type I error have seldom been evaluated. This probably reflects the fact that empirical methods, such as Monte Carlo approaches, are required to assess these concerns. In this study we show that analytical methods cannot always be used to evaluate the robustness of statistical tests, but rather Monte Carlo approaches must be employed. We detail empirical protocols for estimating power and type I error rates for parametric, nonparametric and randomization methods, and demonstrate their application for an analysis of variance and a regression/correlation analysis design. Together, this study provides a framework from which behaviourists can compare the reliability of different methods for data analysis, serving as a basis for selecting the most appropriate statistical test given the characteristics of data at hand. Copyright 2001 The Association for the Study of Animal Behaviour.     

Estimates of a Treatment Effect Based on Censored Data Rank Tests

Linear rank test statistics are applied to the problem of estimating a treatment effect if two sets of censored failure time data are compared and the distributions of the log-failure times of the two samples are assumed to differ only in location. Rank tests for this accelerated failure time model are reviewed and Hodges-Lehmann type estimates for the shift parameter are proposed. Properties of these estimates are investigated, computational aspects are discussed and an example is presented.     

Using Standard Statistics to Consider Uncertainty in Industry-Based Life Cycle Inventory Databases (7 pp)

Goal, Scope and Background Decision-makers demand information about the range of possible outcomes of their actions. Therefore, for developing Life Cycle Assessment (LCA) as a decision-making tool, Life Cycle Inventory (LCI) databases should provide uncertainty information. Approaches for incorporating uncertainty should be selected properly contingent upon the characteristics of the LCI database. For example, in industry-based LCI databases where large amounts of up-to-date process data are collected, statistical methods might be useful for quantifying the uncertainties. However, in practice, there is still a lack of knowledge as to what statistical methods are most effective for obtaining the required parameters. Another concern from the industry's perspective is the confidentiality of the process data. The aim of this paper is to propose a procedure for incorporating uncertainty information with statistical methods in industry-based LCI databases, which at the same time preserves the confidentiality of individual data. Methods The proposed procedure for taking uncertainty in industry-based databases into account has two components: continuous probability distributions fitted to scattering unit process data, and rank order correlation coefficients between inventory flows. The type of probability distribution is selected using statistical methods such as goodness-of-fit statistics or experience based approaches. Parameters of probability distributions are estimated using maximum likelihood estimation. Rank order correlation coefficients are calculated for inventory items in order to preserve data interdependencies. Such probability distributions and rank order correlation coefficients may be used in Monte Carlo simulations in order to quantify uncertainties in LCA results as probability distribution. Results and Discussion A case study is performed on the technology selection of polyethylene terephthalate (PET) chemical recycling systems. Three processes are evaluated based on CO2 reduction compared to the conventional incineration technology. To illustrate the application of the proposed procedure, assumptions were made about the uncertainty of LCI flows. The application of the probability distributions and the rank order correlation coefficient is shown, and a sensitivity analysis is performed. A potential use of the results of the hypothetical case study is discussed. Conclusion and Outlook The case study illustrates how the uncertainty information in LCI databases may be used in LCA. Since the actual scattering unit process data were not available for the case study, the uncertainty distribution of the LCA result is hypothetical. However, the merit of adopting the proposed procedure has been illustrated: more informed decision-making becomes possible, basing the decisions on the significance of the LCA results. With this illustration, the authors hope to encourage both database developers and data suppliers to incorporate uncertainty information in LCI databases.     

The North American obligate cave fauna: regional patterns

The obligate cave faunas of nine regions of the United States –Florida Lime Sinks, Appalachians, Interior Low Plateaus, Ozarks, Driftless Area,Edwards Aquifer/Balcones Escarpment, Guadalupe Mountains, Black Hills, andMother Lode – are described and compared. The number of aquatic(stygobitic) species ranged from zero (Black Hills) to 82 (Appalachians), andthe number of terrestrial (troglobitic) species ranged from zero (Florida LimeSinks) to 256 (Interior Low Plateau). Even at the level of genus, overlapbetween regions is low. Several predictor variables (karst area, number ofcaves, number of long caves, number of deep caves, distance from the Pleistoceneice margin, distance from the late Cretaceous Sea, and vegetation type – asurrogate for productivity) were assessed using rank order statistics,especially rank order multiple regression with a backward elimination procedure.For both stygobites and troglobites, only number of caves was a significantpredictor. The absence of a karst area effect suggested that the degree of karstdevelopment is better described by the number of caves rather than area ofkarst. There was no evidence that distance to Pleistocene glacial boundaries wasimportant, but there was some support for the importance of distance from late Cretaceous sea margins, a potential source of aquatic subterranean colonists. Finally,there was no indication that surface productivity had an effect on speciesrichness. Analysis was complicated by correlations among predictor variables.     

Male Social Behavior and Dominance Hierarchy in the Sulawesi Crested Black Macaque (Macaca nigra)

In a 6-week study of the social behavior of wild Sulawesi crested black macaques (Macaca nigra), we found a linear and transitive dominance hierarchy among the six adult males in one social group. Dominance rank, as determined by the direction of supplantations, correlated strongly with percentage of time near more than four neighbors, frequency of grooming received from adult females, and percentage of time with an adult female as nearest neighbor. These results suggest that high-ranking males are socially attractive. Adult females sexually solicited high-ranking males more often than low-ranking males, but frequency of copulation was not correlated with dominance rank. Frequency and intensity of aggression between males are strongly correlated with rank distance, but aggression toward females was greatest for mid-ranking males. Males of all rank displayed significantly more aggression toward sexually receptive females than toward females in other estrous states. These data indicate that male Sulawesi crested black macaques display a social organization similar to that reported for multimale groups in other macaque species rather than the egalitarian social organization described for female Sulawesi macaques.     

Implication of correlations among some common stability statistics — a Monte Carlo simulations

Stability analysis of multilocation trials is often based on a mixed two-way model. Two stability measures in frequent use are the environmental variance (S _i ² )and the ecovalence (W _i). Under the two-way model the rank orders of the expected values of these two statistics are identical for a given set of genotypes. By contrast, empirical rank correlations among these measures are consistently low. This suggests that the two-way mixed model may not be appropriate for describing real data. To check this hypothesis, a Monte Carlo simulation was conducted. It revealed that the low empirical rank correlation amongS _i ² and W _i is most likely due to sampling errors. It is concluded that the observed low rank correlation does not invalidate the two-way model. The paper also discusses tests for homogeneity of S _i ² as well as implications of the two-way model for the classification of stability statistics.     

Rank Tests for the 2×2 Design with I Blocks

In this paper we suggest rank tests for main effects and for interaction when there are two factors and two levels within each factor and when, in addition, there are I blocks. The asymptotic distributions of the test statistics under H₀ are derived. The application of the tests is illustrated by an example and a simulation study investigates size and power of the tests.     

Comparison of Nonparametric Statistics for Detection of Linkage in Nuclear Families: Single-Marker Evaluation

We have evaluated 23 different statistics, from a total of 10 popular software packages for model-free linkage analysis of nuclear-family data, by applying them to single-marker data simulated under several two-locus disease models. The statistics that we examined fall into two broad categories: (1) those that test directly for increased identity-by-state or identity-by-descent sharing (by use of the programs APM, Genetic Analysis System [GAS] SIBSTATE and SIBDES, SAGE SIBPAL, ERPA, SimIBD, and Genehunter NPL) and (2) those that are based on likelihood-ratio tests and that report LOD scores (by use of the programs Splink, SIBPAIR, Mapmaker/Sibs, ASPEX, and GAS SIBMLS). For each of eight two-locus disease models, we analyzed six data sets; the first three data sets consisted of two-child families with both sibs affected and zero, one, or both parents typed, whereas the other three data sets consisted of four-child families with at least two affected sibs and zero, one, or both parents typed. We report false-positive rates, overall rank by power, and the power for each statistic. We give rough recommendations regarding which programs provide the most powerful tests for linkage, as well as the programs to be avoided under certain conditions. For the likelihood-ratio-based statistics, we examined the effects of various treatments of sibships with multiple affected individuals. Finally, we explored the use of some simple two-of-three composite statistics and found that such tests are of only marginal benefit over the most powerful single statistic.     

Applications of the Mantel-Haenszel statistic to the comparison of survival distributions.

In comparing two survival distributions, a Mantel-Haenszel statistic can be computed after each death as a non-linear two-sample rank statistic. The distributions of both the maximum and terminal statistics in such a sequence are studied numerically, in the absence of censoring, and appropriate critical values are determined. The maximum statistic is applied to simultaneous inference, and both the maximum and terminal statistics are used as the basis for early stopping procedures (especially in the pseudo-sequential context). Procedures based on the two statistics are compared for power and for early decision properties such as stopping index and (for exponential distributions) stopping time.     

A group sequential test for survival trials: an alternative to rank-based procedures

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.     

A powerful and robust test statistic for randomization inference in group-randomized trials with matched pairs of groups

For group-randomized trials, randomization inference based on rank statistics provides robust, exact inference against nonnormal distributions. However, in a matched-pair design, the currently available rank-based statistics lose significant power compared to normal linear mixed model (LMM) test statistics when the LMM is true. In this article, we investigate and develop an optimal test statistic over all statistics in the form of the weighted sum of signed Mann-Whitney-Wilcoxon statistics under certain assumptions. This test is almost as powerful as the LMM even when the LMM is true, but it is much more powerful for heavy tailed distributions. A simulation study is conducted to examine the power.     

Identifying differentially expressed genes from microarray experiments via statistic synthesis

MOTIVATION: A common objective of microarray experiments is the detection of differential gene expression between samples obtained under different conditions. The task of identifying differentially expressed genes consists of two aspects: ranking and selection. Numerous statistics have been proposed to rank genes in order of evidence for differential expression. However, no one statistic is universally optimal and there is seldom any basis or guidance that can direct toward a particular statistic of choice. RESULTS: Our new approach, which addresses both ranking and selection of differentially expressed genes, integrates differing statistics via a distance synthesis scheme. Using a set of (Affymetrix) spike-in datasets, in which differentially expressed genes are known, we demonstrate that our method compares favorably with the best individual statistics, while achieving robustness properties lacked by the individual statistics. We further evaluate performance on one other microarray study.     

A Global View on Parametric and Nonparametric Approaches to the Analysis of Ordered Categorical Data

Rank approaches are very common in the analysis of ordered categorical data but can only be interpreted on an experiment‐wise level. Therefore, parametric tests from linear models, although based on metric structures, are used frequently to analyze this type of data. So the questions arise 1. what parametric tests measure in this context and 2. whether the rank approach could be modified to achieve a global level of interpretation. A possible solution to question 2. offers the so called ridit approach, which is based on known reference distributions. In this paper we discuss a global view that shows how rank analysis and ridit analyses are related and how parametric procedures fit into the same framework. The use of the uniform distribution as a reference in the ridit approach gives an explanation to question 1. The asymptotic multivariate normality of the effect estimators is shown and robust test statistics are discussed. Type I and type II error rates are examined in simulation studies and the approach is applied to a toxicological example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)